Next, we may introduce the group Fourier transform on \(\mathbf {H}_n\)  with
the aid of Schrödinger-type representations, where we follow the notations of the book [1]}. The group Fourier transform for a
function \(f\in L^1(\mathbf {H}_n)\)  is the bounded operator on \(L^2(\mathbb {R}^n)\) , which
can be represented by
\(\pi _{\lambda }(f)=\hat{f}(\lambda ):=\int _{\mathbf {H}_n}f(\eta )\pi _{\lambda }(\eta )^*\mathrm {d}\eta \) 
